[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki5\/analytic-torsion-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki5\/analytic-torsion-wikipedia\/","headline":"Analytic torsion – Wikipedia","name":"Analytic torsion – Wikipedia","description":"Topological invariant of manifolds that can distinguish homotopy-equivalent manifolds In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister\u2013Franz torsion) is a","datePublished":"2018-08-17","dateModified":"2018-08-17","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki5\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki5\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3d6e36e07b9587a556de05b249666876b429c59d","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3d6e36e07b9587a556de05b249666876b429c59d","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki5\/analytic-torsion-wikipedia\/","about":["Wiki"],"wordCount":12835,"articleBody":"Topological invariant of manifolds that can distinguish homotopy-equivalent manifoldsIn mathematics, Reidemeister torsion (or R-torsion, or Reidemeister\u2013Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister (Reidemeister 1935) for 3-manifolds and generalized to higher dimensions by Wolfgang Franz\u00a0(1935) and Georges de Rham\u00a0(1936).Analytic torsion (or Ray\u2013Singer torsion) is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer\u00a0(1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Jeff Cheeger\u00a0(1977, 1979) and Werner M\u00fcller\u00a0(1978) proved Ray and Singer’s conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds.Reidemeister torsion was the first invariant in algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology as a distinct field. It can be used to classify lens spaces.Reidemeister torsion is closely related to Whitehead torsion; see (Milnor 1966). It has also given some important motivation to arithmetic topology; see (Mazur). For more recent work on torsion see the books (Turaev 2002) and (Nicolaescu\u00a02002, 2003).Table of ContentsDefinition of analytic torsion[edit]Definition of Reidemeister torsion[edit]A short history of Reidemeister torsion[edit]Cheeger\u2013M\u00fcller theorem[edit]References[edit]Definition of analytic torsion[edit]If M is a Riemannian manifold and E a vector bundle over M, then there is a Laplacian operator acting on the k-forms with values in E. If the eigenvalues on k-forms are \u03bbj then the zeta function \u03b6k is defined to be"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki5\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki5\/analytic-torsion-wikipedia\/#breadcrumbitem","name":"Analytic torsion – Wikipedia"}}]}]